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56       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(the quotient of G by H) for which the canonical mapping n : G -> G/H is a
homomorphism.

(12.12.1)   Let Gbea topo logicalgroup and Ha normal subgroup ofG. Then the
quotient by H of the topology of G is compatible with the group structure of
G/H.

Let (x0,. y0) be a point of (G/H) x (G/H), and let x0 (resp. yQ) be an
element of the coset x0 (resp. j)0). Every neighborhood of x0y0 is of the form
7c(U), where U is a neighborhood of x0 j>0 . There exists a neighborhood V of
x0 and a neighborhood W of j0 in G such that the relations x e V and
y e W imply xy e U. Hence the relations x e n(V) and y e n(W) imply
xj; e 7r(U), and therefore the mapping (x, y)*-+xy is continuous at the point
(x0, j>0). The proof of continuity of the map xN+x""1 is similar.

Whenever we speak of the quotient G/H of a topological group G by a
subgroup H, it is to be understood that G/H carries the quotient topology.

(12.12.2)   Let Gbea topological group and let H, K be two normal subgroups
of G such that H ID K. Then the canonical bijection (p : G/K -(G/H)/(K/H)
is an isomorphism of topological groups.

We know that (p is the unique mapping which makes the following diagram
commutative:

G-----------> G/K

where n, n', and n" are the canonical homomorphisms. The definition of a
quotient topology (12.10) shows that a subset U of (G/H)/(K/H) is open if
and only if ri~\if-*(\3J) is open in G, that is, if and only if n'l((p"l(U)) is
open in G; but this last condition signifies that ^-1(U) is open in G/K. Since
<p is bijective, the result follows.

(12.12.3) Let Gbea topological group, H a normal subgroup ofG, n: G->G/H
the canonical mapping, and A a subgroup ofG. Then the canonical bijection of
n(A) onto the quotient group AH/H is an isomorphism of topological groups.

This is a particular case of (12.10.7),  applied to the right action
(h, X)H-X/Z of H on G.